PSAT Math Quiz: Rates

What this quiz covers

This quiz focuses on Rates, giving you a quick way to practice the rules, question types, and explanations that matter most for PSAT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $50\text{ gallons}$
2. $65\text{ gallons}$ (correct answer)
3. $78\text{ gallons}$
4. $117\text{ gallons}$

Explanation: We need to find how many gallons the hose fills in 26 minutes at a constant rate. First, calculate the filling rate: 45 gallons ÷ 18 minutes = 2.5 gallons per minute. Then multiply by the new time: 2.5 gallons/minute × 26 minutes = 65 gallons. Common errors include dividing 26 by 45 (which gives a meaningless ratio) or treating 45 as the rate without dividing by 18. Always set up rates as quantity per unit time, then multiply by the desired time.

Question 2

1. $10.5\text{ min}$
2. $12\text{ min}$
3. $14\text{ min}$ (correct answer)
4. $28\text{ min}$

Explanation: We need to find how long it takes to drain 63 gallons when the pump removes 18 gallons every 4 minutes. First, find the drainage rate: 18 gallons ÷ 4 minutes = 4.5 gallons per minute. Then find the time: 63 gallons ÷ 4.5 gallons/minute = 14 minutes. The key is recognizing this as a rate problem where we need gallons per minute first. A common mistake is setting up the proportion backwards, which would give minutes per gallon instead.

Question 3

1. $3.2\text{ cups}$
2. $4.0\text{ cups}$ (correct answer)
3. $6.25\text{ cups}$
4. $8.0\text{ cups}$

Explanation: We need to find how much flour is needed for 32 muffins when 2.5 cups make 20 muffins. First, find the rate of flour per muffin: 2.5 cups ÷ 20 muffins = 0.125 cups per muffin. Then multiply by 32 muffins: 0.125 cups/muffin × 32 muffins = 4.0 cups. This is a scaling problem where we find the unit rate first. Watch out for the temptation to use mental math shortcuts that might introduce rounding errors.

Question 4

1. $3\tfrac{3}{7}$ hr
2. $\tfrac{7}{24}$ hr
3. $\tfrac{24}{7}$ hr (correct answer)
4. 7 hr

Explanation: The question asks for the time to paint 1 room when two workers collaborate. Worker A's rate is 1/6 room per hour, and Worker B's rate is 1/8 room per hour. When working together, add their rates: 1/6 + 1/8 = 4/24 + 3/24 = 7/24 rooms per hour. To find time for 1 room, divide: 1 room ÷ (7/24 rooms/hour) = 1 × 24/7 = 24/7 hours. This equals 3 3/7 hours. The key insight is that rates add when workers collaborate, not times. Always work with rates (rooms per hour) rather than times (hours per room) when combining efforts.

Question 5

1. $900\text{ words}$
2. $1125\text{ words}$ (correct answer)
3. $1350\text{ words}$
4. $1620\text{ words}$

Explanation: We need to find how many words the student types in 25 minutes when typing 540 words in 12 minutes. First, find the typing rate: 540 words ÷ 12 minutes = 45 words per minute. Then multiply by 25 minutes: 45 words/minute × 25 minutes = 1125 words. The key is maintaining consistent units throughout the calculation. A common error is trying to set up a proportion without first finding the unit rate, which can lead to inverted fractions.

Question 6

1. $10\text{ mph}$
2. $12\text{ mph}$
3. $15\text{ mph}$ (correct answer)
4. $20\text{ mph}$

Explanation: We need to find the cyclist's speed represented by the slope of the line through (0,0) and (4,60). The slope equals rise over run: (60-0) miles ÷ (4-0) hours = 60 ÷ 4 = 15 miles per hour. The slope of a distance-time graph always represents speed when distance is on the y-axis and time is on the x-axis. A common error is reversing the division to get 4/60, which would give hours per mile instead of miles per hour. Remember that slope = Δy/Δx, and check your units match what's being asked.

Question 7

1. $20$
2. $24$
3. $30$ (correct answer)
4. $39$

Explanation: This is a classic net rate problem where two processes work in opposite directions. When you encounter scenarios with simultaneous filling and draining, filling and emptying, or similar opposing forces, always calculate the net effect first. The pump adds water at $300$ liters per minute while the leak removes water at $40$ liters per minute. The net rate of water accumulation is $300 - 40 = 260$ liters per minute. To find the time needed to accumulate $7{,}800$ liters, divide the target amount by the net rate: $\frac{7{,}800}{260} = 30$ minutes. Let's examine why the other answers are incorrect. Choice (A) $20$ minutes represents the trap of using only the pump rate: $\frac{7{,}800}{300} = 26$ minutes, which rounds to $20$ if you make calculation errors. Choice (B) $24$ minutes might result from incorrectly adding the rates instead of finding their difference: $\frac{7{,}800}{300 + 40} = \frac{7{,}800}{340} ≈ 23$. Choice (D) $39$ minutes could come from using just the leak rate by mistake: $\frac{7{,}800}{40} = 195$ minutes, though this doesn't directly yield $39$—it's likely a calculation error combined with conceptual confusion. Remember: in rate problems involving opposing forces, always subtract the rates to find the net effect. Don't get distracted by the individual rates—focus on what's actually being accomplished overall. This pattern appears frequently on standardized tests.

Question 8

1. $3.0$
2. $3.6$ (correct answer)
3. $4.5$
4. $5.0$

Explanation: When you encounter work rate problems, think in terms of how much work each person completes per unit of time. This approach turns a seemingly complex scenario into straightforward arithmetic. Painter X completes the job in 6 days, so X's rate is $\frac{1}{6}$ of the house per day. Painter Y completes the job in 9 days, so Y's rate is $\frac{1}{9}$ of the house per day. When working together, you add their rates: $\frac{1}{6} + \frac{1}{9}$. To add these fractions, find a common denominator. The least common multiple of 6 and 9 is 18, so: $\frac{1}{6} = \frac{3}{18}$ and $\frac{1}{9} = \frac{2}{18}$. Their combined rate is $\frac{3}{18} + \frac{2}{18} = \frac{5}{18}$ of the house per day. If they complete $\frac{5}{18}$ of the house per day, then the time to complete the entire house is $\frac{1}{\frac{5}{18}} = \frac{18}{5} = 3.6$ days. This confirms answer choice B. Choice A (3.0) likely comes from averaging the two times incorrectly: $\frac{6+9}{2} = 7.5$, then making additional errors. Choice C (4.5) might result from taking half of 9 days. Choice D (5.0) could come from subtracting 6 from 9, then adding back some arbitrary amount. Remember: in work rate problems, always convert individual completion times to rates (work per unit time), add the rates when people work together, then take the reciprocal to find the total time needed.

Question 9

1. $360\text{ pages}$
2. $450\text{ pages}$
3. $600\text{ pages}$ (correct answer)
4. $720\text{ pages}$

Explanation: When you encounter a combined work rate problem, you need to add the individual rates together to find the total output rate, then multiply by the time period. First, find each printer's rate: Printer A produces 30 pages per minute, and Printer B produces 20 pages per minute. When working together, their combined rate is $30 + 20 = 50$ pages per minute. To find the total pages produced in 12 minutes, multiply the combined rate by the time: $50 \text{ pages/minute} \times 12 \text{ minutes} = 600 \text{ pages}$. This confirms answer choice C is correct. Looking at the wrong answers: Choice A (360 pages) represents a common error where students might calculate only one printer's output over 12 minutes ($30 \times 12 = 360$), forgetting to include the second printer entirely. Choice B (450 pages) could result from incorrectly averaging the two rates first ($\frac{30+20}{2} = 25$), then multiplying by 12 to get $25 \times 12 = 300$, though this doesn't match exactly—it may represent a calculation error along this flawed path. Choice D (720 pages) likely comes from multiplying the rates instead of adding them ($30 \times 20 = 600$), then making an additional error, or from some other computational mistake. Remember: In combined rate problems, always add the individual rates to get the total rate. Think of it as "how much work gets done per unit time when everyone works together." This approach works for any scenario involving multiple workers, machines, or processes operating simultaneously.

Question 10

1. $1.25 per mile
2. $2.00 per mile (correct answer)
3. $3.75 per mile
4. $0.50 per mile

Explanation: This question asks for the driver's charge per mile in dollars per mile, given a base fee and total costs for 6-mile and 10-mile trips in dollars. The total cost follows a linear relationship: cost = base fee + (miles) × (rate in dollars per mile). To find the rate, set up the equations base + 6m = 14.50 and base + 10m = 22.50, then subtract to eliminate the base: 4m = 8, so m = 2 dollars per mile. Unit analysis confirms dollars divided by miles yields dollars per mile, ensuring the rate is properly set up. A common error is dividing total cost by miles without accounting for the base fee, such as 14.50 / 6 ≈ 2.42, which ignores the fixed component. For linear rate problems with a y-intercept, always use the difference in costs over difference in miles to find the slope accurately.

Question 11

1. 9 mi
2. 12 mi
3. 15 mi (correct answer)
4. 18 mi

Explanation: This question asks how many miles a cyclist will travel in 1 hour (60 minutes) at a constant speed, given 4.5 miles in 18 minutes. The speed is constant, so distance = rate (in miles per minute) × time. First, find the rate: 4.5 miles / 18 minutes = 0.25 miles per minute; then for 60 minutes, distance = 0.25 miles/min × 60 min = 15 miles. Unit analysis: (miles per minute) × minutes cancels to miles, stressing proper rate and unit setup. A key error is forgetting to convert hours to minutes, like treating 1 hour as 1 unit without adjustment. For time unit mismatches in rates, convert everything to consistent units like minutes to avoid calculation errors.

Question 12

1. 2.3 hours
2. 3.5 hours (correct answer)
3. 4.0 hours
4. 5.0 hours

Explanation: The question asks how long it will take the cyclist to ride a total of 42 miles, given that 18 miles were already ridden in 1.5 hours at a constant speed, with the answer in hours. The rate relationship is speed in miles per hour, which remains constant throughout the ride. First, calculate the speed: 18 miles ÷ 1.5 hours = 12 miles per hour, ensuring units are miles over hours. The remaining distance is 42 miles - 18 miles = 24 miles, so the time for the remaining distance is 24 miles ÷ 12 miles per hour = 2 hours; adding the initial 1.5 hours gives a total of 3.5 hours. A key error to avoid is misinterpreting 1.5 hours as 1 hour and 50 minutes instead of 1 hour and 30 minutes, which could lead to incorrect speed calculation. Another common mistake is forgetting to subtract the initial distance and calculating time for the full 42 miles. As a test-taking strategy, always double-check unit conversions and what the question is asking for—total time includes the initial ride.

Question 13

1. 20 min
2. 32 min (correct answer)
3. 48 min
4. $\frac{8}{3}$ min

Explanation: The question asks how many minutes it will take to add 80 gallons to the water tank at a constant filling rate. The rate relationship is given by 30 gallons added in 12 minutes, so the filling rate is 30 gallons per 12 minutes or 2.5 gallons per minute after dividing both by 12. To find the time, use time = total gallons / rate, so t = 80 gallons / 2.5 gallons per minute. Calculating that: 80 / 2.5 = 32 minutes, with gallons canceling to leave minutes. Ensure unit consistency by keeping everything in gallons and minutes without unnecessary conversions. A common error is setting up the proportion inversely, like confusing gallons over time with time over gallons, leading to fractions like 8/3. For rate problems, write out the units explicitly in your setup to avoid inversion errors.

Question 14

1. $2.1\text{ hours}$
2. $3.0\text{ hours}$
3. $3.75\text{ hours}$ (correct answer)
4. $4.5\text{ hours}$

Explanation: We need to find how long it takes to travel 126 miles at the same constant speed. First, calculate the van's speed: rate = distance ÷ time = 84 miles ÷ 2.5 hours = 33.6 miles per hour. To find the time for 126 miles, use time = distance ÷ rate = 126 miles ÷ 33.6 mph = 3.75 hours. A common error is to set up the rate as hours per mile (2.5/84) instead of miles per hour, which would give an incorrect answer. When working with rates, always check that your units make sense—speed should be distance per time, not time per distance.

Question 15

1. Machine X, $15\text{ bars/min}$ (correct answer)
2. Machine X, $0.067\text{ min/bar}$
3. Machine Y, $13\text{ bars/min}$
4. Machine Y, $20\text{ bars/min}$

Explanation: We need to compare the packaging rates of two machines in bars per minute. Machine X: 180 bars ÷ 12 minutes = 15 bars/minute. Machine Y: 260 bars ÷ 20 minutes = 13 bars/minute. Since 15 > 13, Machine X has the greater unit rate at 15 bars per minute. A common error is comparing total bars (260 > 180) without accounting for the different time periods. Always convert to the same unit rate before comparing—here, bars per minute makes the comparison straightforward.

Question 16

1. $22.75 (correct answer)
2. $19.50
3. $27.30
4. $18.20

Explanation: We need to find the cost of 3.5 pounds of almonds at the same price per pound. First, find the unit price: $7.80 ÷ 1.2 pounds = $6.50 per pound. Then multiply by the desired amount: $6.50/pound × 3.5 pounds = $22.75. A common mistake is using $7.80 as the price per pound without dividing by 1.2, which would give $27.30. Always calculate the unit rate first by dividing total cost by total quantity before scaling up.

Question 17

1. 90 seconds
2. 360 seconds
3. 450 seconds (correct answer)
4. 540 seconds

Explanation: The question asks how many seconds it will take to fill 90 gallons at the given rate of 12 gallons per minute. The rate is filling speed, in gallons per minute. First, find the time in minutes: 90 gallons / 12 gallons per minute = 7.5 minutes, where gallons cancel to leave minutes. Convert to seconds: 7.5 minutes * 60 seconds per minute = 450 seconds, ensuring unit consistency throughout. Unit analysis is crucial here to handle the conversion from minutes to seconds properly. A common mistake is forgetting the unit conversion, leading to answers in minutes like 7.5, or misapplying it, such as dividing by 60. When rates involve time units, always convert to the requested unit step-by-step to avoid errors.

Question 18

1. $13.00 (correct answer)
2. $11.70
3. $2.60
4. $15.60

Explanation: We need to find the cost of 5 pounds of almonds when 3 pounds cost $7.80. First, find the unit rate: $7.80 ÷ 3 pounds = $2.60 per pound. Then multiply by 5 pounds: $2.60/pound × 5 pounds = $13.00. The key is recognizing this as a unit rate problem where we find dollars per pound first. A common mistake is trying to scale directly without finding the unit rate, which can lead to calculation errors.

Question 19

1. $8.10 (correct answer)
2. $9.00
3. $12.96
4. $4.05

Explanation: This question asks for the cost in dollars of 18 ounces of trail mix at $7.20$ per pound, noting $1$ pound = 16 ounces. The cost is proportional: $cost = \text{pounds} \times \text{rate in dollars per pound}$, requiring ounce-to-pound conversion. Convert 18 ounces to pounds: $18 / 16 = 1.125$ pounds; then $cost = 1.125 \times 7.20 = 8.10$ dollars. Unit analysis: $\frac{\text{ounces}}{\text{ounces per pound}}$ gives pounds, then $\times$ dollars per pound yields dollars, emphasizing unit conversion in rates. A common error is forgetting the conversion, like $18 \times 7.20 / 16$ incorrectly ordered, but here it works if done right. Always convert to the rate's unit first before multiplying to avoid dimensional mismatches.

Question 20

1. $6 per ticket
2. $8 per ticket
3. $9 per ticket (correct answer)
4. $54 per ticket

Explanation: The question asks for the unit rate in dollars per ticket, represented by the slope of the line passing through (0,0) and (6,54). The rate is the slope, in dollars per ticket. Calculate the slope: change in cost / change in tickets = (54 - 0) dollars / (6 - 0) tickets = 9 dollars per ticket. The units confirm dollars divided by tickets give the per-ticket rate. Emphasizing unit analysis helps interpret graph slopes as rates correctly. A key error is using only one coordinate, like 54/6 without considering the origin, but here it's correct; another is misreading the axes. For graph-based rates, always use two points and check if it passes through the origin for proportional relationships.